Math 321  Real Variables II  Spring 2015
Instructor: Malabika Pramanik
Office: 214 Mathematics Building
Email: malabika at math dot ubc dot ca
Lectures: Mon,Wed,Fri 9:00 AM to 10:00 AM in Room 104 Mathematics Building.
Office hours: Mon 1011, Wed 1112 or by appointment.
Course information
Piazza links
Homework
Practice problems
Here is a list of exercises from the textbook based on the weekly lecture material. They are not meant to be turned in for grading, but it is recommended that you work through them to get a better understanding of the topics covered. Some midterm and final exam problems may be modelled on these exercises. Unlike homework problems, detailed solutions of these problems will not be posted, but I am happy to include hints upon request.
 Sequences and series of functions
 Chapter 7: Exercises 126.For problem 22, assume that f is Riemann integrable instead of RiemannStieltjes integrable for now.
 More practice problems on Chapter 7. I will keep updating this list of problems throughout the course as I find new ones.
 RiemannStieltjes integration
 Week 6: Chapter 6, Exercises 17.
 More practice problems on Chapter 6. I will keep updating this list of problems throughout the course as I find new ones.
Real analysis lecture notes on the web
Here is a list of online lecture notes of similar courses offered at various institutions.
MIT open courseware .
John Lindsay Orr's Analysis Webnotes, University of Nebraska, Lincoln.
Eric Sawyer's lecture notes ,McMaster University.
Vern Paulsen's lecture notes, University of Houston.
Lee Larson's lecture notes, University of Louiville.
An introduction to real analysis by William Trench.
Weekbyweek course outline
Here is a rough guideline of the course structure, arranged by week. The textbook pages are mentioned as a reference and as a reading guide. The treatment of these topics in lecture may vary somewhat from that of the text.
 Week 1 (pages 143154 of the textbook):
 Pointwise and uniform convergence
 Definitions and examples
 Interchanging limits.
 Week 2 (pages 143154 of the textbook):
 Applications of pointwise and uniform convergence:
 Weierstrass Mtest
 A continuous but nowhere differentiable function
 A spacefilling curve
 Separability of the space of continuous functions on [0,1]
 Week 3 (pages 143154 of the textbook):
 Density of polygonal functions in the space C[0,1]
 A quick review of uniform continuity
 Bernstein's proof of the Weierstrass approximation theorem
 Week 4 (pages 159161 of the textbook):
 Approximation of continuous periodic functions by trigonometric polynomials
 Algebras and lattices
 The StoneWeierstrass theorem  real version
 Week 5 (pages 154165 of the textbook):
 StoneWeierstrass theorem  complex version (a reading exercise)
 Review of compactness in general metric spaces
 Equicontinuity
 Statement of the ArzelaAscoli theorem
 Week 6 (pages 120130 of the textbook) :
 Proof of the ArzelaAscoli theorem
 Towards RiemannStieltjes integral
 The RiemannStieltjes integral
 Riemann's condition for RiemannStieltjes integrability
 The space of RiemannStieltjes integrable functions
 Week 7 :
 Week 8 :
 Closure under uniform convergence
 Midterm review.
 Week 9 (pages 120133 of the textbook):
 Functions of bounded variation
 Jordan's theorem
 An integral formula for total variation
 Week 10 (pages 120133 of the textbook) :
 Week 11 (pages 120133 of the textbook) :
 Integration by parts
 Integrators of bounded variation
 Riesz representation theorem
 Week 12 (pages 185192 of the textbook):
 Fourier series
 L^2 convergence of Fourier series
 Bessel's inequality
 Plancherel's theorem
 Week 13 (pages 204223 of the textbook):
 Linear transformations and differentiation
 The inverse function theorem
 The implicit function theorem
Worksheets
Midterm

The midterm will be held in class on Friday February 27. The duration of the exam is 50 minutes.
 The syllabus includes the material covered in class up until Monday February 23. In particular, this covers all of Chapter 7 and a portion of Chapter 6 of the textbook.
 Here is the list of problems we discussed in class in Wednesday February 25, with a sketch of the solutions.